Determination of conductivity in anisotropic dipping formations from magnetic coupling measurements

ABSTRACT

A method is disclosed for the determination of horizontal resistivity, vertical resistivity, dip and strike angles of anisotropic earth formations surrounding a wellbore. Electromagnetic couplings among a plural of triad transmitters and triad receivers are measured. Each triad transmitter/receiver consists of coil windings in three mutually orthogonal axes. These measured signals are used to generate initial estimates of the dip angle and strike angle of the formation as well as the anisotropy coefficient and the horizontal resistivity of the formation. An iterative algorithm is then applied using these quantities to finally arrive at more accurate estimates that approach the true values in the formation.

BACKGROUND

1. Field of the Invention

The present invention generally relates to the measurement of electricalcharacteristics of formations surrounding a wellbore. More particularly,the present invention relates to a method for determining horizontal andvertical resistivities in anisotropic formations while accounting forthe dip and stike angle of the formation.

2. Description of the Related Art

The basic principles and techniques for electromagnetic logging forearth formations are well known. Induction logging to determine theresistivity (or its inverse, conductivity) of earth formations adjacenta borehole, for example, has long been a standard and importanttechnique in the search for and recovery of subterranean petroleumdeposits. In brief, the measurements are made by inducing eddy currentsto flow in the formations in response to an AC transmitter signal, andthen measuring the appropriate characteristics of a receiver signalgenerated by the formation eddy currents. The formation propertiesidentified by these signals are then recorded in a log at the surface asa function of the depth of the tool in the borehole.

It is well known that subterranean formations surrounding an earthborehole may be anisotropic with regard to the conduction of electricalcurrents. The phenomenon of electrical anisotropy is generally aconsequence of either microscopic or macroscopic geometry, or acombination thereof, as follows.

In many sedimentary strata, electrical current flows more easily in adirection parallel to the bedding planes, as opposed to a directionperpendicular to the bedding planes. One reason is that a great numberof mineral crystals possess a flat or elongated shape (e.g., mica orkaolin). At the time they were laid down, they naturally took on anorientation parallel to the plane of sedimentation. The interstices inthe formations are, therefore, generally parallel to the bedding plane,and the current is able to easily travel along these interstices whichoften contain electrically conductive mineralized water. Such electricalanisotropy, sometimes called microscopic anisotropy, is observed mostlyin shales.

Subterranean formation are often made up of a series of relatively thinbeds having different lithological characteristics and, thereforedifferent resistivities. In well logging systems, the distances betweenthe electrodes or antennas are great enough that the volume involved ina measurement may include several such thin beds. When individual layersare neither delineated nor resolved by a logging tool, the tool respondsto the formation as if it were a macroscopically anisotropic formation.A thinly laminated sand/shale sequence is a particularly importantexample of a macroscopically anisotropic formation.

If a sample is cut from a subterranean formation, the resistivity of thesample measured with current flowing parallel to the bedding planes iscalled the transverse or horizontal resistivity ρ_(H). The inverse ofρ_(H) is the horizontal conductivity σ_(H). The resistivity of thesample measured with a current flowing perpendicular to the beddingplane is called the longitudinal or vertical resistivity, ρ_(V), and itsinverse the vertical conductivity σ_(V). The anisotropy coefficient λ isdefined as: λ={square root over (σ_(h)+L /σ_(v)+L )}.

In situations where the borehole intersects the formation substantiallyperpendicular to the bedding planes, conventional induction andpropagation well logging tools are sensitive almost exclusively to thehorizontal component of the formation resistivity. When the boreholeintersects the bedding planes at an angle (a deviated borehole) the toolreadings contain an influence from the vertical and horizontalresistivities. This is particularly true when the angle between theborehole and the normal to the bedding places is large, such as indirectional or horizontal drilling, where angles near 90° are commonlyencountered. In these situations, the influence of vertical resistivitycan cause discrepancies between measurements taken in the same formationin nearby vertical wells, thereby preventing a useful comparison ofthese measurements. In addition, since reservoir evaluation is typicallybased on data obtained from vertical wells, the use of data from wellsdrilled at high angles may produce erroneous estimates of formationreserve, producibility, etc. if proper account is not taken of theanisotropy effect.

There have been proposed a number of methods to determine vertical andhorizontal resistivity near a deviated borehole. Hagiwara (U.S. Pat. No.5,966,013) disclosed a method of determining certain anisotropicproperties of formation using propagation tool without a prioriknowledge of the dip angle. In U.S. Pat. No. 5,886,526, Wu described amethod of determining anisotropic properties of anisotropic earthformations using multi-spacing induction tool with assumed functionaldependence between dielectric constants of the formation and itshorizontal and vertical resistivity. Gupta et al. (U.S. Pat. No.5,999,883) utilized a triad induction tool to arrive at an approximateinitial guesses for the anisotropic formation parameters. Moran andGianzero (Geophysics, Vol. 44, P. 1266, 1979) proposed using a tri-axialtool of zero spacing to determine dip angle. Later the spacing wasextended to finite size by Gianzero et al. (U.S. Pat. No. 5,115,198)using a pulsed induction tool. The above references are herebyincorporated herein by reference.

These attempts to determine vertical and horizontal resistivity around adeviated borehole have thus far not provided sufficient accuracy forformations having a high degree of anisotropy. A new technique istherefore needed.

SUMMARY OF THE INVENTION

The above-described problems are in large part addressed by an iterativemethod for determining electrical conductivity in an anisotropic dippingformation. The iterative method corrects for the skin effect to highorders while determining all relevant formation parameters. This methodmay be applied to a tri-axial induction sonde operating in continuouswave (CW) mode. In one embodiment, the method includes (1) measuring amagnetic coupling between transmitter coils and receiver coils of a toolin a borehole traversing the formation; (2) obtaining from the measuredcoupling a strike angle between the tool and the formation; (3)obtaining from the measured coupling an initial dip angle between thetool and the formation; (4) obtaining from the measured coupling aninitial anisotropic factor of the formation; (5) obtaining from themeasured coupling an initial horizontal conductivity of the formation;(6) determining an iterative anisotropic factor from the measuredcoupling, the strike angle, the latest dip angle, and the latestanistropic factor; (7) determining an iterative horizontal conductivityfrom the measured coupling, the strike angle, the latest iterativeanisotropic factor, and the latest dip angle; and (8) determining aniterative dip angle from the measured coupling, the latest iterativeanisotropic factor, and the latest iterative horizontal conductivity.The steps of determining an iterative anisotropic factor, determining aniterative horizontal conductivity, and determining an iterative dipangle are preferably repeated a number of times that minimizes anoverall residual error.

The disclosed method may provide the following advantages in determiningthe formation parameters of anisotropic earth formations: (1) a prioriknowledge of the dip angle is unnecessary and can be one of the outputsof the method; (2) no assumed relationship between formation resistivityand dielectric constant is necessary; (3) complex electronics forpulsing the transmitter coils may be eliminated since this method isapplicable to a triad induction sonde running in CW mode; (4)preliminary results indicate that the disclosed method yields moreaccurate estimates of all electrically relevant formation parameters inthe earth formation.

BRIEF DESCRIPTION OF THE DRAWINGS

A better understanding of the present invention can be obtained when thefollowing detailed description of the preferred embodiment is consideredin conjunction with the following drawings, in which:

FIG. 1 shows the coil configuration of a triazial induction tool;

FIG. 2 demonstrates a rotational transformation definition;

FIG. 3 shows a flow diagram for the disclosed method of determiningformation parameters in a dipping anisotropic earth formation;

FIG. 4 shows a graph used to determine the optimal number of iterationsthat minimize a dip-angle error function; and

FIG. 5 compares the results of the disclosed method to results obtainedwith an existing method.

While the invention is susceptible to various modifications andalternative forms, specific embodiments thereof are shown by way ofexample in the drawings and will herein be described in detail. Itshould be understood, however, that the drawings and detaileddescription thereto are not intended to limit the invention to theparticular form disclosed, but on the contrary, the intention is tocover all modifications, equivalents and alternatives falling within thespirit and scope of the present invention as defined by the appendedclaims.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS Terminology

It is noted that the terms horizontal and vertical as used herein aredefined to be those directions parallel to and perpendicular to thebedding plane, respectively.

Tool Configuration

Turning now to the figures, FIG. 1 shows a conceptual sketch of a coilarrangement for a downhole induction tool. A triad of transmitter coilsT_(x), T_(y) and T_(z), each oriented along a respective axis, isprovided, as is a similarly oriented triad of receiver coils R_(x),R_(y) and R_(z),. For clarity, it is assumed that the three coils ineach triad represent actual coils oriented in mutually perpendiculardirections, with the z-axis corresponding to the long axis of the tool.However, it is noted that this coil arrangement can be “synthesized” byperforming a suitable transformation on differently oriented triads.Such transformations are described in depth in U.S. patent applicationSer. No. 09/255,621 entitled “Directional Resistivity Measurements forAzimutal Proximity Detection of Bed Boundaries” and filed Feb. 22, 1999by T. Hagiwara and H. Song, which is hereby incorporated herein byreference.

Each of the coils in the transmitter triad is parallel to thecorresponding coil in the receiver triad and is spaced away from thecorresponding coil in the z-axis direction. The distance between thecorresponding coils is labeled L. It is noted that the downhole tool mayhave additional transmitter or receiver triads to provide multipletransmitter-receiver spacing L values. Such configurations mayadvantageously provide increased accuracy or additional detail usefulfor analyzing the formation structure.

System Model

Generally, a formation model is used to interpret the tool measurements.The model used herein is a unixial anisotropy model. This model assumesthat the formation is isotropic in the horizontal direction (parallel tothe bedding plane) and anisotropic in the vertical direction(perpendicular to the bedding plane). Setting up a formation coordinatesystem having the z-axis perpendicular to the bedding plane and the x-and y-axes parallel to the bedding plane allows a conductivity tensor tobe expressed as: $\begin{matrix}{\sigma = \begin{bmatrix}\sigma_{h} & 0 & 0 \\0 & \sigma_{h} & 0 \\0 & 0 & \sigma_{v}\end{bmatrix}} & (1)\end{matrix}$

The axes of the formation coordinate system typically do not correspondto the axes of the tool coordinate system. However, a rotationaltransformation from one to the other can be defined. FIG. 2 shows atransformation from the tool coordinate system to the formationcoordinate system. The tool coordinate system (x,y,z) is first rotatedabout the z-axis by an angle β, hereafter termed the strike angle. Theintermediate coordinate system (x′,y′,z′=z) thus formed is then rotatedabout the y′ axis by an angle α, hereafter termed the dip angle toobtain the formation coordinate system (x″,y″=y′,z″).

Any vector v″ in the formation coordinate system can be expressed in thetool coordinate system as:

v=Rv″  (2)

where the rotational transform matrix is: $\begin{matrix}{R = \begin{bmatrix}{\cos \quad \alpha \quad \cos \quad \beta} & {\cos \quad \alpha \quad \sin \quad \beta} & {{- \sin}\quad \alpha} \\{{- \sin}\quad \alpha} & {\cos \quad \beta} & 0 \\{\sin \quad \alpha \quad \cos \quad \beta} & {\sin \quad \alpha \quad \sin \quad \beta} & {\cos \quad \alpha}\end{bmatrix}} & (3)\end{matrix}$

Derivation of Equations

Now that the rotational transformation has been defined, attention isdirected to the induction tool measurements. When a voltage is appliedto one of the transmitter coils, a changing magnetic field is produced.The magnetic field interacts with the formation to induce a voltage inthe receiver coils. Each of the three transmitter coils is excited inturn, and the voltage produced at each of the three receiver coils ismeasured. The nine measured voltages indicate the magnetic couplingbetween the transmitter-receiver triad pair. The equations for themeasured signals will be derived and manipulated to solve for the strikeangle β, the dip angle α, the horizontal conductivity σ_(h), and thevertical anisotropy λ.

In the most general case according to Moran and Gianzero (Geophysics,Vol. 44, P. 1266, 1979), the magnetic field H in the receiver coils canbe represented as a coupling matrix C in the form: $\begin{matrix}{\begin{bmatrix}H_{x} \\H_{y} \\H_{z}\end{bmatrix} = {\begin{bmatrix}C_{xx} & C_{xy} & C_{xz} \\C_{yx} & C_{yy} & C_{zz} \\C_{zx} & C_{zy} & C_{zz}\end{bmatrix}\begin{bmatrix}M_{x} \\M_{y} \\M_{z}\end{bmatrix}}} & (4)\end{matrix}$

where H_(x), H_(y), H_(z) and M_(x), M_(y), M_(z), are the fieldcomponents at the receivers and magnetic moment components at thetransmitters, respectively. (Magnetic moment is calculatedM_(T)=A_(T)N_(T)I_(T), where A_(T) is the transmitter area, N_(T) is thenumber of turns in the transmitter coil, and I_(T) is the transmittercurrent. The direction of the magnetic moment is perpendicular to theplane of the coil). If the coupling matrix is specified in terms of theformation coordinate system, the measured magnetic field strengths inthe receiver coils are obtained by (Moran and Gianzero, Geophysics, Vol.44, P. 1266, 1979):

H=(R ⁻¹ CR)M=C′M  (5)

where H and M are measured in the sonde coordinate system.

Assuming that the tool is oriented so that the strike angle β is 0, itcan be shown that for the uniaxial anisotropy model the full couplingmatrix C′ at the receiver coils (x=0=y, z=L) simplifies to$\begin{matrix}{C^{''} = \begin{bmatrix}C_{xx} & 0 & C_{xz} \\0 & C_{yy} & 0 \\C_{zx} & 0 & C_{zz}\end{bmatrix}} & (6)\end{matrix}$

The theoretical values of the coupling matrix elements are(C_(ij)=C_(ji)): $\begin{matrix}\begin{matrix}{C_{xx} = \quad {\frac{1}{4\pi \quad L^{3}}\lbrack {{k_{h}^{2}L^{2}\cos^{2}{\alpha \cdot ^{{ik}_{h}L}}} - {{ik}_{h}L( \frac{^{{ik}_{h}{LA}} - e^{{ik}_{h}L}}{\sin^{2}\alpha} )} +} }} \\ \quad {( {{3\sin^{2}\alpha} - 1} )( {1 - {{ik}_{h}L}} )^{{ik}_{h}L}} \rbrack\end{matrix} & (7) \\{C_{xy} = {C_{yz} = 0}} & (8) \\{C_{xz} = {\frac{{- \sin}\quad \alpha \quad \cos \quad \alpha}{4\pi \quad L^{3}}{^{{ik}_{h}L}\lbrack {{3( {1 - {{ik}_{h}L}} )} - {k_{h}^{2}L^{2}}} \rbrack}}} & (9) \\\begin{matrix}{C_{yy} = \quad {\frac{1}{4\pi \quad L^{3}}\lbrack {{\frac{k_{h}^{2}L^{2}}{\lambda \quad A^{2}}^{{ik}_{h}L}} + {{ik}_{h}L( \frac{^{{ik}_{h}{LA}} - ^{{ik}_{h}L}}{\sin^{2}\alpha} )} -} }} \\ \quad {( {1 - {{ik}_{h}L}} )^{{ik}_{h}L}} \rbrack\end{matrix} & (10) \\{C_{zz} = {\frac{1}{4\pi \quad L^{3}}{^{{ik}_{h}L}\lbrack {{k_{h}^{2}L^{2}\sin^{2}\alpha} + {( {1 - {{ik}_{h}L}} )( {{3\quad \cos^{2}} - \alpha - 1} )}} \rbrack}}} & (11)\end{matrix}$

where

k_(h)={square root over (ωμσ_(h)+L )}=horizontal wave number

ω=2πƒ=angular frequency

μ=μ₀=4π×10⁻⁷ henry/m=magnetic permeability

λ={square root over (σ_(h)+L /σ_(v)+L )}=anisotropy coefficient

A={square root over (sin² +L α+λ² +L cos² +L α)}/λ=anisotropic factor

In terms of the elements of the coupling matrix C′, the six independentmeasurements for all the possible couplings between alltransmitter-receiver pairs are expressed as (T_(i)R_(j)=T_(j)R_(i)):

T_(x)R_(x)={fraction (M/2+L )}{[(c_(xx)+c_(zz))+(c_(xx)−C_(zz))cos2α+2C_(xz) sin 2α]cos² β+2C_(yy) sin² β}  (12)

T_(y)R_(y)={fraction (M/2+L )}{[(C_(xx)+C_(zz))+(C_(xx)−C_(zz))cos2α+2C_(xz) sin 2α]sin² β+2C_(yy) cos² β}  (13)

T_(z)R_(z)={fraction (M/2+L )}[(C_(xx)+C_(zz))+(C_(zz)−C_(xx))cos2α−2C_(xz) sin 2α]  (14)

T_(x)R_(y)={fraction (M/4+L )}[(C_(xx)+C_(zz))+(C_(xx)−C_(zz))cos2α+2C_(xz) sin 2α−2C_(yy)]sin 2β  (15)

T_(z)R_(x)={fraction (M/2+L )}[(C_(zz)+C_(xx))sin 2α+2C_(xz) cos 2α]cosβ  (16)

T_(z)R_(y)={fraction (M/2+L )}[(C_(zz)−C_(xx))sin 2α+2C_(xz) cos 2α]sinβ  (17)

These measurements are made by taking the ratio of the transmit andreceive voltage signals, e.g. T_(x)R_(y)=K V_(Ry)/V_(Tx), where K is areal-valued calibration constant theoretically equal toA_(T)N_(T)I_(T)A_(R)N_(R)(ωμ)²/4πL, where A_(R) is the area of thereceive coil, and N_(R) is the number of turns of the receive coil.

Explicitly solving the last four of the above equations results in thefollowing expressions for the measured cross-coupling fields:$\begin{matrix}\begin{matrix}{{T_{x}R_{y}} = \quad {\frac{M\quad \sin \quad 2\beta}{8\pi \quad L^{3}}\lbrack {{k_{h}^{2}{L^{2}( {^{{ik}_{h}L} - \frac{^{{ik}_{h}{LA}}}{\lambda^{2}A}} )}} -} }} \\ \quad {{ik}_{h}{L( {^{{ik}_{h}{LA}} - ^{{ik}_{h}L}} )}\frac{( {1 + {\cos^{2}\alpha}} )}{\sin^{2}\alpha}} \rbrack\end{matrix} & \text{(18-a)} \\{{T_{z}R_{x}} = {\frac{M\quad \cos \quad \beta \quad \sin \quad 2\alpha}{8\pi \quad L^{3}\sin^{2}\alpha}{ik}_{h}{L( {^{{ik}_{h}{LA}} - ^{{ik}_{h}L}} )}}} & \text{(18-b)} \\{{T_{z}R_{y}} = {\frac{M\quad \sin \quad \beta \quad \sin \quad 2\alpha}{8\pi \quad L^{3}\sin^{2}\alpha}{ik}_{h}{L( {^{{ik}_{h}{LA}} - ^{{ik}_{h}L}} )}}} & \text{(18-c)} \\{{T_{z}R_{z}} = {\frac{M}{4\pi \quad L^{3}}\lbrack {{2{^{{ik}_{h}L}( {1 - {{ik}_{h}L}} )}} - {{ik}_{h}{L( {^{{ik}_{h}{LA}} - ^{{ik}_{h}L}} )}}} \rbrack}} & \text{(18-d)}\end{matrix}$

To make practical use of the above equations, the real component isignored and the imaginary (reactive) component is simplified by findingthe limit as the transmitter-receiver spacing approaches zero, i.e.,L→0. Doing this simplifies the reactive components of the measuredsignal equations (18-a,b,c) to: $\begin{matrix} ( {T_{x}R_{y}} )_{x}arrow{\frac{M\quad \sin \quad 2\beta \quad \sin \quad \alpha}{8\pi \quad \lambda^{2}\delta_{h}^{3}}( {1 - \lambda^{2}} )}  & \text{(19-a)} \\ ( {T_{z}R_{x}} )_{x}arrow{\frac{M\quad \cos \quad \beta \quad \sin \quad 2\alpha}{8\pi \quad \lambda^{2}\delta_{h}^{3}}( {1 - \lambda^{2}} )}  & \text{(19-b)} \\ ( {T_{z}R_{y}} )_{x}arrow{\frac{M\quad \sin \quad \beta \quad \sin \quad 2\quad \alpha}{8\pi \quad \lambda^{2}\delta_{h}^{3}}( {1 - \lambda^{2}} )}  & \text{(19-c)}\end{matrix}$

where δ_(h) ={square root over (2+L /ωμσ_(h)+L )} is the skin depthassociated with horizontal conductivity. From these equations, onearrives at the practical equations for the determination of dip andstrike angles: $\begin{matrix}{\beta = {\tan^{- 1}\lbrack \frac{( {T_{z}R_{y}} )_{x}}{( {T_{z}R_{x}} )_{x}} \rbrack}} & (20) \\{\alpha_{a1} = {\tan^{- 1}\lbrack \sqrt{\lbrack \frac{( {T_{x}R_{y}} )_{x}}{( {T_{z}R_{x}} )_{x}} \rbrack^{2} + \lbrack \frac{( {T_{x}R_{y}} )_{x}}{( {T_{z}R_{y}} )_{x}} \rbrack^{2}} \rbrack}} & (21)\end{matrix}$

It is noted that the strike angle β thus obtained is exact while the dipangle α is only an approximation because Equations (19a-c) are validonly in the zero-spacing limit. The subscript α1 denotes that this isthe first approximation of the apparent dip angle.

With the strike angle β and the estimated dip angle α, estimates of thehorizontal conductivity σ_(h) and anisotropy factor A can be obtainedvia the following observation.

When a power series expansion is used for the exponential terms inequation (18-b), the first terms yield the following expressions for thereal part (T_(z)R_(x))_(R) and imaginary part (T_(z)R_(x))_(X).$\begin{matrix}{( {T_{z}R_{x}} )_{R} \approx {\frac{M}{8\pi \quad L^{3}}\frac{\cos \quad \beta \quad \sin \quad 2\alpha}{\sin^{2}\alpha}( {A^{2} - 1} )( \frac{L}{\delta_{h}} )^{3}}} & (22) \\{( {T_{z}R_{x}} )_{X} \approx {\frac{M}{8\pi \quad L^{3}}{\frac{\cos \quad \beta \quad \sin \quad 2\alpha}{\sin^{2}\alpha}\lbrack {{{- 2}( {A - 1} )( \frac{L}{\delta_{h}} )^{2}} + {( {A^{2} - 1} )( \frac{L}{\delta_{h}} )^{3}}} \rbrack}}} & (23)\end{matrix}$

Taking advantage of the fact that the second term in (T_(z)R_(x))_(X) isidentical to (T_(z)R_(x))_(R), an equation that is skin-effect correctedto the first order can be written as: $\begin{matrix}{{( {T_{z}R_{x}} )_{R} - ( {T_{z}R_{x}} )} \approx {\frac{M}{4\pi \quad L^{3}}\frac{\cos \quad \beta \quad \sin \quad 2\alpha}{\sin^{2}\alpha}( {A - 1} )( \frac{L}{\delta_{h}} )^{2}}} & (24)\end{matrix}$

Similarly with equation (18-d), for T_(z)R_(z) one has: $\begin{matrix}{{( {T_{z}R_{z}} )_{R} - ( {T_{z}R_{z}} )_{X}} \approx {\frac{M}{4\pi \quad L^{3}}{A( \frac{L}{\delta_{h}} )}^{2}}} & (25)\end{matrix}$

Equations (24) and (25) can be solved to obtain A and σ_(h):$\begin{matrix}{A \approx \frac{1}{1 + {\frac{( {T_{z}R_{x}} )_{X} - ( {T_{z}R_{x}} )_{R}}{( {T_{z}R_{z}} )_{X} - ( {T_{z}R_{z}} )_{R}} \cdot \frac{\tan \quad \alpha}{\cos \quad \beta}}}} & (26) \\{\sigma_{h} \approx {\frac{4\pi \quad L}{M\quad \omega \quad \mu}\frac{\lbrack {( {T_{z}R_{x}} )_{X} - ( {T_{z}R_{x}} )_{R}} \rbrack}{A}}} & (27)\end{matrix}$

Substituting the strike angle β and the first estimate of the dip angleα_(α1) from Equations (20) and (21) yields the first estimates of theanisotropic factor A_(α1) and the horizontal conductivity σ_(h1), wherethe subscript α1 denotes the quantities are the first estimates of theapparent values: $\begin{matrix}{A_{a1} \approx \frac{1}{1 + {\frac{( {T_{z}R_{x}} )_{X} - ( {T_{z}R_{x}} )_{R}}{( {T_{z}R_{z}} )_{X} - ( {T_{z}R_{z}} )_{R}} \cdot \frac{\tan \quad \alpha_{a1}}{\cos \quad \beta}}}} & (28) \\{\sigma_{h1} \approx {\frac{4\pi \quad L}{M\quad \omega \quad \mu}\frac{\lbrack {( {T_{z}R_{x}} )_{X} - ( {T_{z}R_{x}} )_{R}} \rbrack}{A_{a1}}}} & (29)\end{matrix}$

It is noted that the leading fraction may be replaced by a calibrationcoefficient K₁ for the tool.

Now that initial estimates have been obtained, the estimates can beiteratively improved. Examining the series expansion of all the measuredfields reveals that $\begin{matrix}{( {T_{z}R_{x}} )_{R} = {\frac{M}{8\pi \quad L^{3}}\frac{\cos \quad \beta \quad {\sin ( {2\alpha} )}}{\sin^{2}\alpha}\begin{Bmatrix}{{( {A^{2} - 1} )( \frac{L}{\delta_{h}} )^{3}} + {\frac{2}{3}( {1 - A^{3}} )( \frac{L}{\delta_{h}} )^{4}} +} \\{{\frac{1}{6}( {A^{4} - 1} )( \frac{L}{\delta_{h}} )^{5}} + {O\lbrack ( {L/\delta_{h}} )^{6} \rbrack}}\end{Bmatrix}}} & (30) \\{( {T_{z}R_{x}} )_{X} = {\frac{M}{8\pi \quad L^{3}}\frac{\cos \quad \beta \quad {\sin ( {2\alpha} )}}{\sin^{2}\alpha}\begin{Bmatrix}{{2( {1 - A} )( \frac{L}{\delta_{h}} )^{2}} + {( {A^{2} - 1} )( \frac{L}{\delta_{h}} )^{3}} -} \\{{\frac{1}{6}( {A^{4} - 1} )( \frac{L}{\delta_{h}} )^{5}} + {O\lbrack ( {L/\delta_{h}} )^{6} \rbrack}}\end{Bmatrix}}} & (31)\end{matrix}$

Because the higher order terms do not cancel each other, Equations (24)and (25) are no longer correct. To remedy this problem, the extraskin-effect contribution may be subtracted from the left-hand side.Namely, the left-hand side of Equation (24) may be replaced by(T_(z)R_(x))_(R)−(T_(z)R_(x))_(X)−Γ_(zx) where Γ_(zx) is the higherorder correction obtained from (30): $\begin{matrix}{\Gamma_{zx} = {\frac{M}{8\pi \quad L^{3}}{\frac{\cos \quad \beta \quad \sin \quad 2\quad \alpha_{a1}}{\sin^{2}\alpha_{a1}}\lbrack {{\frac{2}{3}( {1 - A_{a1}^{3}} )( \frac{L}{\delta_{a1}} )^{4}} + {\frac{1}{3}( {1 - A_{a1}^{4}} )( \frac{L}{\delta_{a1}} )^{5}}} \rbrack}}} & (32)\end{matrix}$

Similarly, the correction for (T_(z)R_(z))_(R)−(T_(z)R_(z))_(X) can bederived: $\begin{matrix}{\Gamma_{zz} = {\frac{M}{4\pi \quad L^{3}} \cdot \lbrack {{{- \frac{1}{3}}( {1 + {2A_{a1}^{3}}} )( \frac{L}{\delta_{a1}} )^{4}} + {\frac{( {3 + {5A_{a1}^{2}}} )}{15}( \frac{L}{\delta_{a1}} )^{5}}} \rbrack}} & (33)\end{matrix}$

It is noted that the leading fraction in (32) and (33) may be replacedwith a calibration coefficient K₂ and 2K₂, respectively, for the tool.With these corrections, one gets better approximations for A and σ_(h):$\begin{matrix}{A_{a2} = \frac{1}{1 + {\frac{( {T_{z}R_{x}} )_{X} - ( {T_{z}R_{x}} )_{R} - \Gamma_{zx}}{( {T_{z}R_{z}} )_{X} - ( {T_{z}R_{z}} )_{R} - \Gamma_{zz}} \cdot \frac{\tan \quad \alpha_{a1}}{\cos \quad \beta}}}} & (34) \\{\sigma_{h2} = {\frac{4\pi \quad L}{M\quad \omega \quad \mu}\frac{\lbrack {( {T_{z}R_{x}} )_{X} - ( {T_{z}R_{x}} )_{R}} \rbrack}{A_{a2}}}} & (35)\end{matrix}$

The same “extend to a higher order” procedure can be applied to theother components of the measured fields in Equation (25). The end resultis a more accurate equation for the dip angle: $\begin{matrix}{{\tan \quad \alpha_{a}} = \sqrt{\lbrack \frac{( {T_{x}R_{y}} )_{R} - ( \Delta_{xy} )_{R}}{( {T_{z}R_{x}} )_{R} - ( \Delta_{zx} )_{R}} \rbrack^{2} + \lbrack \frac{( {T_{x}R_{y}} )_{R} - ( \Delta_{xy} )_{R}}{( {T_{z}R_{y}} )_{R} - ( \Delta_{zy} )_{R}} \rbrack^{2}}} & (36)\end{matrix}$

It is noted that here the real part of the magnetic field (which isidentical to the imaginary part of the measured voltages at the coilsother than a constant factor) is used, hence the subscript R. Thecorrection terms can be directly obtained from the 6^(th) orderexpansions of the corresponding coupling fields. The correction termsare: $\begin{matrix}{( \Delta_{zx} )_{R} = {\frac{M}{8\pi \quad L^{3}}{\frac{\cos \quad \beta \quad {\sin ( {2\alpha_{a}} )}}{\sin^{2}\alpha_{a}}\lbrack {{\frac{2}{3}( {1 - A_{a}^{3}} )( \frac{L}{\delta_{ha}} )^{4}} + {\frac{1}{6}( {A_{a}^{4} - 1} )( \frac{L}{\delta_{ha}} )^{5}}} \rbrack}}} & \text{(37-a)} \\{( \Delta_{zy} )_{R} = {\frac{M}{8\pi \quad L^{3}}{\frac{\sin \quad \beta \quad {\sin ( {2\alpha_{a}} )}}{\sin^{2}\alpha_{a}}\lbrack {{\frac{2}{3}( {1 - A_{a}^{3}} )( \frac{L}{\delta_{ha}} )^{4}} + {\frac{1}{6}( {A_{a}^{4} - 1} )( \frac{L}{\delta_{ha}} )^{5}}} \rbrack}}} & \text{(37-b)} \\{( \Delta_{xy} )_{R} = {\frac{M\quad \sin \quad 2\beta}{2\pi \quad L^{3}}\lbrack {{{- {P( \alpha_{a} )}}( \frac{L}{\delta_{ha}} )^{4}} + {{Q( \alpha_{a} )}( \frac{L}{\delta_{ha}} )^{5}}} \rbrack}} & \text{(37-c)}\end{matrix}$

where

α_(α)=Apparent dip angle$\delta_{ha} = {\sqrt{\frac{2}{{\varpi\mu\sigma}_{ha}}} = {{Apparent}\quad {horizontal}\quad {skin}\text{-}{depth}}}$

σ_(hα)=Apparent horizontal conductivity${P( \alpha_{a} )} = {{\frac{( {A_{a}^{3} - 1} )}{6}\frac{( {1 + {\cos^{2}\alpha_{a}}} )}{\sin^{2}\alpha_{a}}} + {\frac{A_{a}}{2}\frac{( {A_{a}^{2} - {\cos^{2}\alpha_{a}}} )}{\sin^{2}\alpha_{a}}} - \frac{1}{2}}$${Q( \alpha_{a} )} = {{\frac{( {A_{a}^{4} - 1} )}{24}\frac{( {1 + {\cos^{2}\alpha_{a}}} )}{\sin^{2}\alpha_{a}}} + {\frac{A_{a}^{2}}{6}\frac{( {A_{a}^{2} - {\cos^{2}\alpha_{a}}} )}{\sin^{2}\alpha_{a}}} - \frac{1}{6}}$

Property Determination Method

The above derivation has provided:

Eqn. (20) for the determination of the strike angle β,

Eqn. (21) for a first estimate of the dip angle α,

Eqn. (28) for a first estimate of the anisotropic factor A,

Eqn. (29) for a first estimate of the horizontal conductivity σ,

Eqn. (34) for an iterative estimate of the anisotropic factor A,

Eqn. (35) for an iterative estimate of the horizontal conductivity σ,and

Eqn. (36) for an iterative estimate of the dip angle α.

Eqn. (36) includes correction terms that are specified by Eqn. (37). Theproperty determination approach built into Eqns (34)-(37) is believed tobe unknown to the art.

FIG. 3 shows an iterative method that uses the above equations todetermine the dip and strike angles, the horizontal conductivity, andthe anisotropy coefficient (calculated from the anisotropic factor). Inblock 302, the transmitter coils are activated and the voltages producedin the receiver coils are measured. The measured voltages are processedto determine the magnetic couplings between the coils. In block 304, thestrike angle β is calculated per Eqn (20) and the first estimate of thedip angle α_(A1) is calculated per Eqn (21). In block 306, the firstestimate of the anisotropic factor A_(A) is calculated per Eqn (28) andthe first estimate of the horizontal conductivity σ_(hA) is calculatedper Eqn (29).

Blocks 308 through 314 form a loop having a loop index J. Loop index Jis initialized to 1 for the first iteration of the loop, and incrementedby 1 for subsequent iterations of the loop.

In block 308, an iterative estimate of the anisotropic factor A_(J) iscalculated per Eqn (34) and an iterative estimate of the horizontalconductivity σ_(hJ) is calculated per Eqn (35). In block 310, couplingcorrection terms are caculated per Eqns (37-a,b,c). In block 312, aniterative estimate of the dip angle α_((J+1)) is calculated per Eqn (36)using the coupling correction terms found in block 310. In block 314, atest is made to determine if the optimum number of iterations has beenperformed, and if more iterations are needed, control returns to block308. Otherwise. in block 316, the anisotropy coefficient λ is calculatedfrom the latest estimates of the dip angle and anisotropic factor (referto the definition of the anisotropic factor in Eqn (11)). The anisotropycoefficient λ is then output, along with the strike angle β, and thebest estimates of the dip angle α and horizontal conductivity σ_(h).

Analysis of Method

The effectiveness of the method is demonstrated in Table I where allapparent formation parameters are listed for each successive iteration.The true formation parameters are: dip angle α=45°, strike angle β=60°,anisotropy coefficient λ=2, and horizontal conductivity σ_(h)=5 S/m. Thesonde is a triad 2C40 (notation means 2 coils spaced 40 inches apart)operating at 8 kHz.

TABLE I Effect of iteration number for triad 2C40 in a dippinganisotropic formation. Iteration No. Apparent Dip α_(α) Apparent λApparent σ_(hα) 0 34.72° 4.71 9.234 1 39.21° 1.916 4.752 2 41.74° 1.9684.890 3 43.22° 1.992 4.975 4 44.11° 2.005 5.029 5 44.65° 2.013 5.060 644.99° 2.017 5.083 7 45.19° 2.020 5.095 8 45.32° 2.021 5.104 9 45.40°2.022 5.109 True Values 45.00° 2.000 5.000

A close examination of Table I reveals that there exists an optimaliteration number for each formation parameter beyond which the accuracyof apparent formation parameter actually decreases. For a given triadsonde, the optimal number of iterations for a given parameter can becalculated with the following method. For the three formationparameters, their corresponding error functions are defined as:$\begin{matrix}{ɛ_{\alpha} = {\sum\limits_{\alpha}{\sum\limits_{\sigma_{h}}{\sum\limits_{\lambda}( \frac{\alpha - \alpha_{a}}{\alpha} )^{2}}}}} & \text{(38-a)} \\{ɛ_{\lambda} = {\sum\limits_{\alpha}{\sum\limits_{\sigma_{h}}{\sum\limits_{\lambda}( \frac{\lambda - \lambda_{a}}{\lambda} )^{2}}}}} & \text{(38-b)} \\{ɛ_{\sigma_{h}} = {\sum\limits_{\alpha}{\sum\limits_{\sigma_{h}}{\sum\limits_{\lambda}( \frac{\sigma_{h} - \sigma_{ha}}{\sigma_{a}} )^{2}}}}} & \text{(38-c)}\end{matrix}$

The summations are performed over the range of expected formation dipangles, conductivities, and anisotropy coefficients. This error functionis consequently indicative of the overall residual error.

Given a configuration of a triad sonde, the functional dependence of theerror function on the number of iterations can be examined. FIG. 4 showsthe apparent dip angle examination results for four triad-pairconfigurations having different transmitter-receiver spacings. The errorfunction is calculated over the following ranges of formationparameters:

α=(5, 15, 30, 45, 60, 75, 85°)

σ_(h)=(0.001, 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10 S/m)

λ=(1.1, 1.414, 2, 4, 6, 8, 10).

The optimal iteration number is defined as the iteration number at whichthe error is minimal. This can similarly be done for the anisotropycoefficient λ and horizontal conductivity σ_(h) to obtain theirrespective optimal iteration numbers.

With the optimal iteration number determined, the advantage of thecurrent invention in the dip angle determination over an existing method(Moran and Gianzero, Geophysics, Vol. 44, P. 1266, 1979) can bedemonstrated. FIG. 5 plots the calculated dip angle measured in dippingformations having different horizontal conductivities. The true dipangle of the formation is 75 degrees. It is noted that the originalmethod proposed by Moran and Gianzero has large errors in conductiveformations due to the skin effect, while the disclosed method yields acalculated dip angle close to the true dip of 75 degrees. It is furtheremphasized that in addition to a more accurate dip angle, the currentmethod also yields the other formation parameters illustrated in TableI.

Numerous variations and modifications will become apparent to thoseskilled in the art once the above disclosure is fully appreciated. It isintended that the following claims be interpreted to embrace all suchvariations and modifications.

What is claimed is:
 1. A method for determining conductivity in aformation, wherein the method comprises: measuring a magnetic couplingbetween transmitter coils and receiver coils of a tool in a boreholetraversing the formation; obtaining from the measured coupling a strikeangle between the tool and the formation; obtaining from the measuredcoupling an initial dip angle between the tool and the formation;obtaining from the measured coupling an initial anisotropic factor ofthe formation; obtaining from the measured coupling an initialhorizontal conductivity of the formation; determining an iterativeanisotropic factor from the measured coupling, the strike angle, thelatest dip angle, and the latest anistropic factor; determining aniterative horizontal conductivity from the measured coupling, the strikeangle, the latest iterative anisotropic factor, and the latest dipangle; and determining an iterative dip angle from the measuredcoupling, the latest iterative anisotropic factor, and the latestiterative horizontal conductivity.
 2. The method of claim 1, furthercomprising: repeating the steps of determining an iterative anisotropicfactor, determining an iterative horizontal conductivity, anddetermining an iterative dip angle.
 3. The method of claim 2, whereinsaid repeating is performed a number of times that minimizes an overallresidual error.
 4. The method of claim 1, wherein the strike angle βcorresponds to${\beta = {\tan^{- 1}\lbrack \frac{( {T_{z}R_{y}} )_{x}}{( {T_{z}R_{x}} )_{x\quad}} \rbrack}},$

wherein (T_(z)R_(y))_(X) is the reactive component of the couplingT_(z)R_(y) between a transmitter T_(z) oriented along a z-axis and areceiver R_(y) oriented along a y-axis, and (T_(z)R_(x))_(X) is thereactive component of the coupling T_(z)R_(x) between transmitter T_(z)and a receiver R_(x) oriented along an x-axis.
 5. The method of claim 1,wherein the initial dip angle α₁ corresponds to${\alpha_{1} = {\tan^{- 1}\lbrack \sqrt{\lbrack \frac{( {T_{x}R_{y}} )_{x}}{( {T_{z}R_{x}} )_{x}} \rbrack^{2} + \lbrack \frac{( {T_{x}R_{y}} )_{x}}{( {T_{z}R_{y}} )_{x}} \rbrack^{2}} \rbrack}},$

wherein (T_(x)R_(y))_(X) is the reactive component of the couplingT_(x)R_(y) between a transmitter T_(x) oriented along an x-axis and areceiver R_(y) oriented along a y-axis, (T_(z)R_(x))_(X) is the reactivecomponent of the coupling T_(z)R_(x) between a transmitter T_(z)oriented along a z-axis and a receiver R_(x) oriented along the x-axis,and (T_(z)R_(y))_(X) is the reactive component of the couplingT_(z)R_(y) between transmitter T_(z) and receiver R_(y).
 6. The methodof claim 1, wherein the initial anisotropic factor A₁ corresponds to${A_{1} = \frac{1}{1 + {\frac{( {T_{z}R_{x}} )_{X} - ( {T_{z}R_{x}} )_{R}}{( {T_{z}R_{z}} )_{X} - ( {T_{z}R_{z}} )_{R}} \cdot \frac{\tan \quad \alpha_{1}}{\cos \quad \beta}}}},$

wherein (T_(z)R_(x))_(X) and (T_(z)R_(x))_(R) are the imaginary and realcomponents, respectively, of the coupling T_(z)R_(x) between atransmitter T_(z) oriented along a z-axis and a receiver R_(x) orientedalong an x-axis, (T_(z)R_(z))_(X) and (T_(z)R_(z))_(R) are the imaginaryand real components, respectively, of the coupling T_(z)R_(z) betweentransmitter T_(z) and a receiver R_(z) oriented along the z-axis, α₁ isthe initial dip angle, and β is the strike angle.
 7. The method of claim1, wherein the initial horizontal conductivity σ_(h1) corresponds to${\sigma_{h1} = {K_{1}\frac{\lbrack {( {T_{z}R_{x}} )_{X} - ( {T_{z}R_{x}} )_{R}} \rbrack}{A_{1}}}},$

wherein (T_(z)R_(x))_(X) and (T_(z)R_(x))_(R) are the imaginary and realcomponents, respectively, of the coupling T_(z)R_(x) between atransmitter T_(z) oriented along a z-axis and a receiver R_(x) orientedalong an x-axis, A₁ is the initial anisotropic factor, and K₁ is apredetermined function of transmitter signal voltage and frequency. 8.The method of claim 1, wherein the iterative anisotropic factor A_(i+1)corresponds to${A_{i + 1} = \frac{1}{1 + {\frac{( {T_{z}R_{x}} )_{X} - ( {T_{z}R_{x}} )_{R} - \Gamma_{zx}}{( {T_{z}R_{z}} )_{X} - ( {T_{z}R_{z}} )_{R} - \Gamma_{zz}} \cdot \frac{\tan \quad \alpha_{i}}{\cos \quad \beta}}}},$

wherein (T_(z)R_(x))_(X) and (T_(z)R_(x))_(R) are the imaginary and realcomponents, respectively, of the coupling T_(z)R_(x) between atransmitter T_(z) oriented along a z-axis and a receiver R_(x) orientedalong an x-axis, (T_(z)R_(z))_(X) and (T_(z)R_(z))_(R) are the imaginaryand real components, respectively, of the coupling T_(z)R_(z) betweentransmitter T_(z) and a receiver R_(z) oriented along the z-axis, α_(i)is the latest dip angle, β is the strike angle, Γ_(zx) is a firstskin-effect correction, and Γ_(zz) is a second skin effect correction.9. The method of claim 8, wherein the first skin-effect correctionΓ_(zx) corresponds to$\Gamma_{zx} = {K_{2}{\frac{\cos \quad \beta \quad \sin \quad 2\quad \alpha_{i}}{\sin^{2}\alpha_{i}}\lbrack {{\frac{2}{3}( {1 - A_{i}^{3}} )( \frac{L}{\delta_{hi}} )^{4}} + {\frac{1}{3}( {1 - A_{i}^{4}} )( \frac{L}{\delta_{hi}} )^{5}}} \rbrack}}$

and the second skin-effect correction corresponds to${\Gamma_{zz} = {2{K_{2} \cdot \lbrack {{{- \frac{1}{3}}( {1 + {2A_{i}^{3}}} )( \frac{L}{\delta_{hi}} )^{4}} + {\frac{( {3 + {5A_{i}^{2}}} )}{15}( \frac{L}{\delta_{hi}} )^{5}}} \rbrack}}},$

wherein L is a distance between the transmitter and receiver, A_(i) isthe latest anistropic factor, δ_(hi) ={square root over (2+L /ωμσ_(hi)+L)} is the latest skin depth, σ _(hi) is the latest horizontalconductivity, and K₂ is a predetermined function of transmitter signalvoltage.
 10. The method of claim 1, wherein the iterative horizontalconductivity σ_(h(i−1)) corresponds to${\sigma_{h{({i + 1})}} = {K_{1}\frac{\lbrack {( {T_{z}R_{x}} )_{X} - ( {T_{z}R_{z}} )_{R}} \rbrack}{A_{i + 1}}}},$

wherein (T_(z)R_(x))_(X) and (T_(z)R_(x))_(R) are the imaginary and realcomponents, respectively, of the coupling T_(z)R_(x) between atransmitter T_(z) oriented along a z-axis and a receiver R_(x) orientedalong an x-axis, A_(i−1) is the latest anisotropic factor, L is adistance between the transmitter and receiver, and K₁ is a predeterminedfunction of transmitter signal voltage and frequency.
 11. The method ofclaim 1, wherein the iterative dip angle α_(i+1) corresponds to${\alpha_{i + 1} = {\tan^{- 1}( \sqrt{\lbrack \frac{( {T_{x}R_{y}} )_{R} - ( \Delta_{xy} )_{R}}{( {T_{z}R_{x}} )_{R} - ( \Delta_{zx} )_{R}} \rbrack^{2} + \lbrack \frac{( {T_{x}R_{y}} )_{R} - ( \Delta_{xy} )_{R}}{( {T_{z}R_{y}} )_{R} - ( \Delta_{zy} )_{R}} \rbrack^{2}} )}},$

wherein (T_(x)R_(y))_(R) is the real component of the couplingT_(x)R_(y) between a transmitter T_(x) oriented along an x-axis and areceiver R_(y) oriented along a y-axis, (T_(z)R_(x))_(R) is the realcomponent of the coupling T_(z)R_(x) between a transmitter T_(z)oriented along a z-axis and a receiver R_(x) oriented along the x-axis,(T_(z)R_(y))_(R) is the real component of the coupling T_(z)R_(y)between transmitter T_(z) and receiver R_(y), (Δ_(xy))_(R) is a firstcorrection term, (Δ_(zx))_(R) is a second correction term, and(Δ_(zy))_(R) is a third correction term.
 12. The method of claim 11,wherein the first correction term (Δxy)_(R) corresponds to${( \Delta_{xy} )_{R} = {\frac{M\quad \sin \quad 2\beta}{2\quad \pi \quad L^{3}}\lbrack {{{- {P( \alpha_{i} )}}( \frac{L}{\delta_{h{({i + 1})}}} )^{4}} + {{Q( \alpha_{i} )}( \frac{L}{\delta_{h{({i + 1})}}} )^{5}}} \rbrack}},{with}$${{P( \alpha_{i} )} = {{\frac{( {A_{i + 1}^{3} - 1} )}{6}\frac{( {1 + {\cos^{2}\alpha_{i}}} )}{\sin^{2}\alpha_{i}}} + {\frac{A_{i + 1}^{2}}{6}\frac{( {A_{i + 1}^{2} - {\cos^{2}\alpha_{i}}} )}{\sin^{2}\alpha_{i}}} - \frac{1}{2}}},{and}$${{Q( \alpha_{i} )} = {{\frac{( {A_{i + 1}^{4} - 1} )}{24}\frac{( {1 + {\cos^{2}\alpha_{i}}} )}{\sin^{2}\alpha_{i}}} + {\frac{A_{i + 1}^{2}}{6}\frac{( {A_{i + 1}^{2} - {\cos^{2}\alpha_{i}}} )}{\sin^{2}\alpha_{i}}} - \frac{1}{6}}},$

wherein the second correction term (Δ_(zx))_(R) corresponds to${( \Delta_{zx} )_{R} = {\frac{M}{8\pi \quad L^{3}}{\frac{\cos \quad \beta \quad {\sin ( {2\alpha_{i}} )}}{\sin^{2}\alpha_{i}}\lbrack {{\frac{2}{3}( {1 - A_{i + 1}^{3}} )( \frac{L}{\delta_{h{({i + 1})}}} )^{4}} + {\frac{1}{6}( {A_{i + 1}^{4} - 1} )( \frac{L}{\delta_{h{({i + 1})}}} )^{5}}} \rbrack}}},$

and wherein the third correction term (Δzy)_(R) corresponds to${( \Delta_{zy} )_{R} = {\frac{M}{8\pi \quad L^{3}}{\frac{\sin \quad \beta \quad {\sin ( {2\alpha_{i}} )}}{\sin^{2}\alpha_{i}}\lbrack {{\frac{2}{3}( {1 - A_{i + 1}^{3}} )( \frac{L}{\delta_{h{({i - 1})}}} )^{4}} + {\frac{1}{6}( {A_{i + 1}^{4} - 1} )( \frac{L}{\delta_{h{({i + 1})}}} )^{5}}} \rbrack}}},$

wherein α_(i) is the latest dip angle, β is the strike angle, L is adistance between the transmitter and receiver, A_(i−1) is the latestanistropic factor, δ_(h(i+)1)={square root over (2+L /ωμσ_(h(i+1))+L )}is the latest skin depth, σ _(h(i+1)) is the latest horizontalconductivity, and K₂ is a predetermined function of transmitter signalvoltage.
 13. The method of claim 1, wherein the transmitter coilsconsist of a triad of mutually orthogonal transmitters.
 14. The methodof claim 13, wherein the receiver coils consist of a triad of mutuallyorthogonal receivers.
 15. The method of claim 1, wherein said measuringincludes exciting each transmitter coil in turn and measuring in-phaseand quadrature phase voltage signals induced in each of the receivercoils by each of the transmitter coils.
 16. A method for determiningconductivity in a formation, wherein the method comprises: receivingmagnetic coupling measurements from an induction tool; obtaining fromthe magnetic coupling measurements an initial horizontal conductivity ofthe formation; determining an iterative horizontal conductivity from themagnetic coupling measurements and the initial horizontal conductivityof the formation, wherein the initial horizontal conductivity σ_(h1)corresponds to${\sigma_{h1} = {K_{1}\frac{\lbrack {( {T_{z}R_{x}} )_{X} - ( {T_{z}R_{x}} )_{R}} \rbrack}{A_{1}}}},$

wherein K₁ is a predetermined function of transmitter signal voltage andfrequency, (T_(z)R_(x))_(X) and (T_(z)R_(x))_(R) are the imaginary andreal components, respectively, of the magnetic coupling measurementT_(z)R_(x) between a transmitter T_(z) oriented along a z-axis and areceiver R_(x) oriented along an x-axis, and A₁ is the initialanisotropic factor corresponding to${A_{1} = \frac{1}{1 + {\frac{( {T_{z}R_{x}} )_{X} - ( {T_{z}R_{x}} )_{R}}{( {T_{z}R_{z}} )_{X} - ( {T_{z}R_{z}} )_{R}} \cdot \frac{\tan \quad \alpha_{1}}{\cos \quad \beta}}}},$

wherein the strike angle β corresponds to${\beta = {\tan^{- 1}\lbrack \frac{( {T_{z}R_{y}} )_{x}}{( {T_{z}R_{x}} )_{x}} \rbrack}},$

and the initial dip angle α₁ corresponds to${\alpha_{1} = {\tan^{- 1}\lbrack \sqrt{\lbrack \frac{( {T_{x}R_{y}} )_{x}}{( {T_{z}R_{x}} )_{x}} \rbrack^{2} + \lbrack \frac{( {T_{x}R_{y}} )_{x}}{( {T_{z}R_{y}} )_{x}} \rbrack^{2}} \rbrack}},$

wherein (T_(z)R_(z))_(X) and (T_(z)R_(z))_(R) are the imaginary and realcomponents, respectively, of the coupling T_(z)R_(z) between transmitterT_(z) and a receiver R_(z) oriented along the z-axis, wherein(T_(x)R_(y))_(X) is the reactive component of the coupling T_(x)R_(y)between a transmitter T_(x) oriented along the x-axis and a receiverR_(y) oriented along a y-axis, wherein (T_(z)R_(y))_(X) is the reactivecomponent of the coupling T_(z)R_(y) between transmitter T_(z) andreceiver R_(y).
 17. A method for determining conductivity in aformation, wherein the method comprises: receiving magnetic couplingmeasurements from an induction tool; obtaining from the magneticcoupling measurements an initial horizontal conductivity of theformation; determining an iterative horizontal conductivity from themagnetic coupling measurements and the initial horizontal conductivityof the formation, wherein the iterative horizontal conductivityσ_(h(i+1)) corresponds to${\sigma_{h{({i + 1})}} = {K_{1}\frac{\lbrack {( {T_{z}R_{x}} )_{X} - ( {T_{z}R_{x}} )_{R}} \rbrack}{A_{i + 1}}}},$

wherein K₁ is a predetermined function of transmitter signal voltage andfrequency, wherein (T_(z)R_(x))_(X) and (T_(z)R_(x))_(R) are theimaginary and real components, respectively, of the coupling T_(z)R_(x)between a transmitter T_(z) oriented along a z-axis and a receiver R_(x)oriented along an x-axis, and A_(i+1) corresponds to${A_{i + 1} = \frac{1}{1 + {\frac{( {T_{z}R_{x}} )_{X} - ( {T_{z}R_{x}} )_{R} - \Gamma_{zx}}{( {T_{z}R_{z}} )_{X} - ( {T_{z}R_{z}} ) - \Gamma_{zz}} \cdot \frac{\tan \quad \alpha_{i}}{\cos \quad \beta}}}},$

wherein (T_(z)R_(z))_(X) and (T_(z)R_(z))_(R) are the imaginary and realcomponents, respectively, of the coupling T_(z)R_(z) between transmitterT_(z) and a receiver R_(z) oriented along the z-axis, Γ_(zx) is a firstskin-effect correction, Γ_(zz) is a second skin effect correction, thestrike angle β corresponds to${\beta = {\tan^{- 1}\lbrack \frac{( {T_{z}R_{y}} )_{x}}{( {T_{z}R_{x}} )_{x}} \rbrack}},$

and the dip angle α_(i) corresponds to${\alpha_{i} = {\tan^{- 1}( \sqrt{\lbrack \frac{( {T_{x}R_{y}} )_{R} - ( \Delta_{xy} )_{R}}{( {T_{z}R_{x}} )_{R} - ( \Delta_{zx} )_{R}} \rbrack^{2} + \lbrack \frac{( {T_{x}R_{y}} )_{R} - ( \Delta_{xy} )_{R}}{( {T_{z}R_{y}} )_{R} - ( \Delta_{zy} )_{R}} \rbrack^{2}} )}},$

wherein (T_(z)R_(y))_(X) and (T_(z)R_(y))_(R) are the imaginary and realcomponents, respectively, of the coupling T_(z)R_(y) between transmitterT_(z) and a receiver R_(y) oriented along a y-axis, (T_(x)R_(y))_(R) isthe real component of the coupling T_(x)R_(y) between a transmitterT_(x) oriented along the x-axis and receiver R_(y), (Δ_(xy))_(R) is afirst correction term, (Δ_(zx))_(R) is a second correction term, and(Δ_(zy))_(R) is a third correction term.
 18. The method of claim 17,wherein the first skin-effect correction Γ_(zx) corresponds to$\Gamma_{zx} = {K_{2}{\frac{\cos \quad {\beta sin}\quad \alpha_{i}}{\sin^{2}\alpha_{i}}\lbrack {{\frac{2}{3}( {1 - A_{i}^{3}} )( \frac{L}{\delta_{hi}} )^{4}} + {\frac{1}{3}( {1 - A_{i}^{4}} )( \frac{L}{\delta_{hi}} )^{5}}} \rbrack}}$

and the second skin-effect correction corresponds to${\Gamma_{zz} = {2{K_{2} \cdot \lbrack {{{- \frac{1}{3}}( {1 + {2A_{i}^{3}}} )( \frac{L}{\delta_{hi}} )^{4}} + {\frac{( {3 + {5A_{i}^{2}}} )}{15}( \frac{L}{\delta_{hi}} )^{5}}} \rbrack}}},$

wherein L is a distance between the transmitter and receiver, A_(i) isthe latest anistropic factor, δ_(hi) ={square root over (2/ωμσ_(hi)+L )}is the latest skin depth, σ _(hi) is the latest horizontal conductivity,and K₂ is a predetermined function of transmitter signal voltage. 19.The method of claim 17, wherein the first correction term (Δ_(xy))_(R)corresponds to${( \Delta_{xy} )_{R} = {\frac{M\quad \sin \quad 2\beta}{2\quad \pi \quad L^{3}}\lbrack {{{- {P( \alpha_{i - 1} )}}( \frac{L}{\delta_{hi}} )^{4}} + {{Q( \alpha_{i - 1} )}( \frac{L}{\delta_{hi}} )^{5}}} \rbrack}},{with}$${{P( \alpha_{i} )} = {{\frac{( {A_{i + 1}^{3} - 1} )}{6}\frac{( {1 + {\cos^{2}\alpha_{i}}} )}{\sin^{2}\alpha_{i}}} + {\frac{A_{i + 1}}{6}\frac{( {A_{i + 1}^{2} - {\cos^{2}\alpha_{i}}} )}{\sin^{2}\alpha_{i}}} - \frac{1}{2}}},{and}$${{Q( \alpha_{i} )} = {{\frac{( {A_{i + 1}^{4} - 1} )}{24}\frac{( {1 + {\cos^{2}\alpha_{i}}} )}{\sin^{2}\alpha_{i}}} + {\frac{A_{i + 1}^{2}}{6}\frac{( {A_{i + 1}^{2} - {\cos^{2}\alpha_{i}}} )}{\sin^{2}\alpha_{i}}} - \frac{1}{6}}},$

wherein the second correction term (Δzx)_(R) corresponds to${( \Delta_{zx} )_{R} = {\frac{M}{8\pi \quad L^{3}}{\frac{\cos \quad \beta \quad {\sin ( {2\alpha_{i - 1}} )}}{\sin^{2}\alpha_{i - 1}}\lbrack {{\frac{2}{3}( {1 - A_{i}^{3}} )( \frac{L}{\delta_{hi}} )^{4}} + {\frac{1}{6}( {A_{i}^{4} - 1} )( \frac{L}{\delta_{hi}} )^{5}}} \rbrack}}},$

and wherein the third correction term (Δzy)_(R) corresponds to${( \Delta_{zy} )_{R} = {\frac{M}{8\pi \quad L^{3}}{\frac{\sin \quad \beta \quad {\sin ( {2\alpha_{i - 1}} )}}{\sin^{2}\alpha_{i - 1}}\lbrack {{\frac{2}{3}( {1 - A_{i}^{3}} )( \frac{L}{\delta_{hi}} )^{4}} + {\frac{1}{6}( {A_{i}^{4} - 1} )( \frac{L}{\delta_{hi}} )^{5}}} \rbrack}}},$

wherein α_(i) is the latest dip angle, β is the strike angle, L is adistance between the transmitter and receiver, A_(i) is the latestanistropic factor, δ_(hi) ={square root over (2+L /ωμσ_(hi)+L )} is thelatest skin depth, σ _(hi) is the latest horizontal conductivity, K₂ isa predetermined function of transmitter signal voltage.